We solve problem (REF ) using the backward induction approach outlined in [1]}. We assume \(L<\infty \) , but this is without loss of generality. Notice that (REF ) can be rewritten as
\(V_l^L=\sup _{l \le \tau \le L} \mathbb {E}_{(X,Y)\sim {D}}\left[\gamma ^{(\tau )}\right],\) 
